1a. Assume that the stocks traded in the stock exchange are A, B, ..., Z (i.e., the alphabet). In addition, consider that Amy was the sole person involved in the selection of the good portfolio before the announcement made by CGFA. In particular, she examined through programming all possible sets of four different stocks so as to calculate portfolio returns and standard deviations. What is the probability that she ended up choosing the GOLD Portfolio (note: for the sake of simplicity, please assume that all stocks have the same characteristics, and that the stocks were equally weighted in each set)? 1b. From the probability theory viewpoint, what are the two other necessary assumptions that lead to your answer in 1a? How would your answer in 1a change if we considered that all founders contributed to the selection of the GOLD Portfolio (note: an answer without computations is enough)?

In light of such a predictable goal (i.e., wealth creation), they announce the creation of a valuable, but risky portfolio: the GOLD portfolio. Now that the we know that CGFA will market the GOLD Portfolio, the following information becomes relevant.

Risk-Free Rate 0.05%

Economy Condition Probability Stock G Stock O Stock L Stock D

Catastrophic 0.80% -63.25% -31.25% -8.20% -0.50%

Bad 31.20% -12.70% -9.55% -4.50% -0.10%

Neutral 38.35% 1.10% 0% 0.75% 0.40%

Good 29.35% 9.55% 12.15% 1.60% 0.50%

Exuberant 0.30% 80.45% 75.35% 12.50% 3.10%

Capital Allocations 13.50% 34.70% 8.05% 43.75%

1c. In light of the information above, compute the expected values and standard deviations of stocks G, O, L, and D.

1d. By considering the stocks separately, which one should be chosen for investment purposes? Why?

1e. In light of the information above, calculate the expected return and standard deviation of the GOLD Portfolio (hint: while both ways are doable, it may be more interesting to work with a new random variable rather than many covariance terms...).

Thanks to their knowledge in behavioural finance, the four founders of CGFA could establish their utility functions

Founder Utility Function

Amy U(A) = E[r] + 0.97 ²

Brian U(B) = E[r] - 0.15 [exp(4 - 1) -1]

Chris U(C) = E[r] + 0.65 [ln(7 + 5) - 1]

Daisy U(D) = E[r] - 1.00 (3)

In addition, they decided that it would be prudent to work with a combination of the GOLD Portfolio and the risk-free asset: the CGFA Portfolio.

1f. In light of their rookie level, the four founders of CGFA agreed that it is better for now that the management of the CGFA Portfolio be conducted by manager(s) with an exclusive risk-averse profile. In that sense, who would you choose to manage the capital of the clients of CGFA? Please explain the reasoning that led to your choice(s).

1g. Based on your choice(s) and that the fact that the maximum aditional stress level (i.e., maximum marginal loss of utility) that the manager(s) is(are) willing to endure is dU/d = -1.13, please compute the capital allocation in the GOLD Portolio, and the expected return and standard deviation of the CGFA Portfolio.